“First Baptist” simply means that a particular church was the “first” Baptist church established in an area. The next oldest Baptist church then is entitled to use the name “Second Baptist” if it so wishes. “Southern Baptist” refers to a convention of Baptist churches particularly prominent in the US South.
As for Primitive Baptists, they are very rigid Baptists reacting to what they perceive to be the mass departure from Baptist distinctives from such groups as the Southern Baptists and other mainstream Baptists.
They are called Primitive because of their rejection of organized, centralized denominationalized missionary efforts. In the eighteenth and nineteenth centuries, many Baptist churches began to cooperate together for missionary work and building institutions like seminaries. Groups like the American Baptist Churches USA and the Southern Baptist Convention are the results of such efforts. These Baptists were called “missionary Baptists” because they supported organized missionary efforts among Baptist churches.
The Primitive Baptists were the anti-missionary Baptists because they opposed centralized missionary agencies because they felt that these would try to control the local Baptist congregations, which was contrary to their beliefs about the autonomy of the local church.